Part 9

NOTE.--I thought that I could not bring before the audience the character of this decoration adequately, except by showing some of the designs, and some of the furniture, on a screen. Some of the pictures were taken from Niccolini’s magnificent _Art of Pompeii_ (Naples, 1876-92), the Curator having allowed me to use the expensive process of photographing in colours, in order to show not only the design, but the rich colours of the Pompeian walls.

VI

SCIENCE: GRAMMAR--LOGIC--MATHEMATICS--MEDICINE

When I speak to you of Greek Science, of course I use the word in the old and proper sense to include all strict reasoning, especially of the deductive kind, particularly therefore pure Mathematics, and not merely the inferences from observation and experiment which now commonly assume and even monopolise the title of Science. I often see in educational programmes Science and Mathematics contrasted as distinct things, which indeed in this case they are, only because the Science so-called is often unworthy of the name. Sciences of observation were, I think, not formulated by the Greeks except in the case of Medicine, in which their results are still quoted with respect; in the case of Hydrostatics, as Heron’s great book shows; and in the case of Natural History, in which they made the first collection of facts that modern men of science can use; but we have lost what they said on their artistic observations, namely their minute observations of the anatomy of the human body, which, as I have told you, their sculptors learned to represent with such accuracy that no modern anatomist can find a flaw in their work. This was done by careful external observation, for the practice of dissecting the human body would have seemed to them impious and horrible. But, whenever it was possible, the Greeks went back to first principles and framed a theory from which they deduced the facts; and this it is which has made their science so valuable. It will not be hard to show you how in Logic--the Science of Reasoning,--in Arithmetic, and in Geometry--the science of the laws of lines, of figures, and of solid bodies in space--they are our teachers to the present day.

It is well to approach the subject of Logic through the avenue by which the Greeks approached it, through the analysis of ordinary language and as the natural expression of thinking. The early poets and great prose writers had so far perfected the use of language that the Greeks in the catalogue of human acquisitions came to put their speech on a very high pedestal. Delighted with it, and despising all other tongues as barbarous, they convinced themselves that the Greek word adequately expressed the nature of the thing it signified, and therefore that to understand their language properly was to understand the nature of things. Λὁγος meant not only speech (_oratio_), but reason (_ratio_), and so, after first seeking to obtain clear conceptions of abstract ideas, they advanced to the structure of sentences and analysed speech in so accurate a way that their technical terms are our technical terms of to-day. When you talk of _infinitives_, or _genitives_, or _participles_, you are only using words borrowed from Latin translations, often mistranslations, of the Greek. You find these logical studies in their beginning, but by no means in their infancy, in the _Dialogues_ of Plato. Whole conversations are employed in trying to fix the connotation of important moral terms, such as _holiness_, or _valour_, or _temperance_. And we also find in some of the dialogues an appreciation of the difficulties contained in the form of simple propositions, the meaning of affirmation or negation, and the nature of the deduction of one proposition from another.

But I need not detain you with particulars about these early preparations for science, when we have before us in Aristotle various treatises on the analysis of speech from its logical side, and the laying down of the laws of formal thinking with such accuracy and completeness that nothing of importance has ever been added to it. We hear it often said that a single man apprehended and systematised these laws. That is not true; there were plenty of tentative essays before his time. But if there be one achievement which has made his name and fame everlasting, it is his treatment of the theory of Reasoning.

The mediæval universities knew this well, and so do the modern universities of Europe which are worthy of the name. I need not bear witness to the vast importance of common Logic by telling you that in my own youth nothing ever woke me up like having a good Logic put into my hands at the age of fourteen. For since that time I have been often teaching it and have watched its effects on hundreds of intelligent youths. Among all the subjects that we teach, not for the purpose of supplying mere facts, but for the purpose of training youth to judge facts and co-ordinate their knowledge, I know nothing that benefits the average student like the study of Aristotelian Logic. May I add that, so far as I know American education, the most serious defect I have observed in it is the small attention paid to this subject, and hence the vast number of your men and women who are unable to distinguish a sound from an unsound argument, still less to point out where the fallacy lies.

There are here present, I have no doubt, a large number of people, otherwise highly educated, who, were I to propose a stock example for their criticism, would feel at a loss how to deal with it. Let me give an illustration. “Every hen comes from an egg; every egg comes from a hen; therefore every egg comes from an egg.” Is this a correct argument? If not, where is it at fault? If you had all been trained in Whately’s Logic, or any other Logic of the kind, as we were in our youth, such a question would present no difficulty whatever.

But if you have failed to derive this lesson from the old Greeks, your English ancestors were better advised. All the subtlety of the mediæval schools, all the disputations of their universities, were based on Greek Logic; and, if they often wasted their time on idle problems, it must always be remembered that by this means Europe was trained to accuracy and subtlety in argument, and hence to weigh vague and random theorising and to make men competent critics of any new dogma. We often remark from our side of the Atlantic how many wild theories in religion, how many sham theories in science, blossom and flourish in this country, inhabited though it may be by a most shrewd and intelligent population. The simplest answer is to point to their ignorance of common Logic, and hence their liability to be deceived by the most vulgar fallacies. It would be easy to mention a book popular in this country, the pages of which any logically trained people would only use to wrap sardines or to heat a stove.

The Greeks do not parade their logic in their writings, though we know they were fond of subtleties; there are indeed examples of it in the _Sophist_ of Plato, where this sort of thing is ridiculed in his travesty of two professional educators. But there are two great and solid proofs of the power which strict Logic had upon their minds. The first comes out in their literature. Wherever they undertake to argue an issue, whether political, social, or religious, their reasoning is clear and easily followed. They of course often start from traditional beliefs, which may not now command assent, but they always reason from these with clear and sober thinking. There was no more important cause for the permanence of that great literature. Its sound thinking has kept it from all extravagance and made it acceptable to educated men of all ages and nations. The second proof is my chief subject to-day: it is the peculiarly logical character of Greek mathematics which has made this too the model of the scientific thinking of the world.

Let me go back to the infancy of Greek science and give you evidence for this statement. Setting aside for the present the metaphysical thinkers, who will occupy us in another chapter, we may safely say that the earliest mathematicians were the school of Pythagoras, and also that their work started (so far as they did not start from the highest of all--pure thinking) from Arithmetic. To this science Pythagoras and his school attached such importance that they were supposed to hold that numbers were the essence of the universe. If you think that such a theory is mere nonsense, I may tell you that I have often heard my colleagues, distinguished in modern science, discuss a theory, alive at the present day, that the so-called material universe consists of mere motion, without anything to be moved! At the root of these speculations lies the fundamental distinction of form and matter, of the definite and the indefinite; and the Pythagoreans had got a glimpse of the eternal truth that it is only through our intuitions of space and time, and through abstract concepts explicating these, that we can bring the myriad phenomena of nature under intelligible law. It was an early anticipation, so far as we can explain it, of the great theory of Descartes, that all the universe could be reduced to mathematical relations, and these handled by algebra, which is in its essence but a very abstract and generalised arithmetic. If therefore all parts of the world stand in mutual arithmetical relations, of which the chemical law of definite proportions is the most signal example, the science of numbers must be the capital of every scientific man.

And remember that in Greek parlance this was the strict meaning of their _arithmetic_--a pure science, while they used the term _logistic_ (or computation) for the working of practical rules. At the basis of their theory of numbers lay of course the one great assumption which makes the science possible--I mean the absolute equality of the units of any number used for the purpose of calculation.

This is not merely the abstraction from all their differences, as when I say that the present audience consists of five hundred people, regardless of the countless variations existing between the units of this crowd. It is the assumption of an ideal and accurate identity between each of the units, as to magnitude, which makes the expression of geometrical truths arithmetically possible.

The truth that 3² + 4² = 5² applies not only to numbers but to lines, and probably suggested the geometrical proof to Euclid (1, 47). But it is only true if the units in the measurement of each line are exactly equal.

Starting from this first assumption, the Pythagoreans began to speculate on the peculiarities of the natural series of units in use among men, and to deduce from these general considerations various theorems, which they believed might solve the secrets of nature. At the very outset they were struck with the obvious contrast between odd and even, which Plato, following them, regarded as a fundamental distinction in nature. Had they been told that, thousands of years later, men of science would find that a most primitive and fundamental distinction among animals is founded on this difference, I mean that of _artio-dactyle_, and _perisso-dactyle_, actually called by the Greek words, they would have said that this caused them no surprise, as their arithmetic had long since laid down the distinction as a law of nature. As simple specimens of the sort of treatment that the science of numbers received from them, I may cite the following: The successive additions of the odd numbers produce the squares of the series of even and odd.[32] The series of even numbers when added give us no such result, but rather this--that the addition of even numbers gives us figures which are the products of successive numbers differing by only one, _e.g._ 2 + 4 = 3 × 2; 2 + 4 + 6 = 4 × 3, and so on. These latter numbers were regarded as rectangles, when expressed in lines. It was by the discovery of the relation of the sides to the base of a right-angled triangle that they, so to speak, stumbled upon irrational numbers. If the two sides are each equal to 1, the hypothenuse is equal to √2, which is no integral number, but a problem in itself.[33]

All the results of this Pythagorean research lived through into the days of Plato and Aristotle and then, as we know from Euclid and Theon, into the learning of Alexandria. The importance recognised by them in the numbers ten and twelve was shown by the general adoption of a decimal system of notation, and of the division of time on a duodecimal system.

You will ask me what symbols the Greeks had which could enable them to treat arithmetical figures of any complexity, and on this I could give you now a very definite reply, but the details would lead us away from our subject, seeing that this notation was lost in the Dark Ages and was ultimately replaced by the Arabic numerals. But we now know that they had a very practical system of decimal notation based on the use of the letters of the alphabet; and the fact that several letters obsolete in the alphabet of the fifth century B.C. appear as symbols, proves that it was current as early as Pythagorean days. The sign for 6 is the _digamma_, that for 90 is the _koph_ of the Phœnician alphabet, which is still found in Locrian inscriptions; the Phœnician letter known as _sampi_ is used for 900. We know the practical management of this easy notation perfectly from the mass of accounts both private and public found on Egyptian papyri. It can express large numbers far more compendiously than the Roman system, often more compendiously even than ours. Suppose you desire to express any large number, say 20,050, here it is β/ΜΝ; say 47,678, it is δ/ΜΖΧΟΗ, and if there be small gain in simplicity here, I will give you 800,000 = 10,000 × 80 = π/Μ. But these are practical matters, though without an easy notation even the most scientific thinkers could not make large progress.[34]

The next great step was to pass from arithmetic to geometry as the science of space and to show how far the same laws governed both.

If we are not well informed upon the beginnings of arithmetic, we are more fortunate in the case of geometry, and here, if anywhere, the old Greeks have been the acknowledged teachers of modern Europe. For we have in the so-called _Elements_ of Euclid, composed most probably at Alexandria about 300 B.C., a summary of all that had been discovered up to his day, doubtless with many new things of his own. He had distinctly built upon his predecessors; he has before him all through his book a problem discussed in Plato, that of the possible number of regular polyhedra, and its solution forms the climax of his work. But he begins from the very beginning and builds up his whole doctrine with such accuracy that a flaw in the demonstration is hard to be found.

How did this great master attain to such perfection? The form of his demonstrations does not suggest an intimacy with the logic of his immediate predecessor Aristotle; but from him he might easily have obtained the whole notion of a strictly deductive science, which, starting from the smallest possible number of primary data, proceeds to derive from these by strict demonstration proposition after proposition. Philosophers of our own time have often expressed wonder at the clearness with which these data are laid down. They are three in kind: first the _common notions_, which apply to all science and all practical life, such as “the whole is greater than its part”; secondly, the _axioms_ peculiar to our intuition of space, such as “two right lines cannot enclose a space”; and thirdly the very simple _postulates_, which amount to the use of a ruler and compass with a pencil. There are besides very careful definitions, so careful that they are at first obscure, because they apply to the ideal construction of the mind in its intuition of pure space and do not concern themselves about the flaws of actual figures. Thus his “point which has no parts” is not nothing at all, but the minimum of definite place; his right line, “which lies in the same way (όμαλῶς) between any two points taken upon its length,” is simply unity of direction. Every other line varies in direction in some of its successive parts. This is a direct appeal to intuition, without which we can make no beginning in the science of space. Such also is the axiom about parallel lines. Such is also the proof by superposition, to show that two triangles, if some of their measurements be the same, must wholly coincide.

But I must not attempt to give you a lecture on the _Elements_ of Euclid, of which some of you may have evil recollections. For it is the misfortune, as well as the glory, of a great work not only to be repeated for centuries, but to be parroted and travestied by those who merely accept its greatness from the voice of ages, and who come to think that the words of inspiration only require blind repetition to instruct men. So if Euclid has become in many classical schools a sort of amulet or fetish (which must for common decency be put in the programme but which may be learned by committing the proofs to memory without any intelligence) such a misfortune is not the fault of Euclid, but the most pathetic tribute to his genius. Let me also add, for the benefit of those of you who have never seen more than six books of the _Elements_, and who probably thought six more than enough, that these are but the introduction to the discussion of higher and more complex questions, which show the large advance made by the Greeks in this science, and which explain also how in other arts, such as architecture, there is no defect for want of scientific accuracy. Books VII-X are not on geometry, but on higher arithmetic, and even treat, as in Book X, of incommensurable or irrational quantities. With XI he begins to teach solid geometry, the measurement of pyramids, cones, spheres, and the like, ending (XIII) with the discussion of the five regular polyhedra, of which Plato had long since spoken.

From the great sequence of discoverers and teachers of pure mathematics, I need only here pick out three immortal names: Apollonius of Perga, living about 200 B.C., whose geometrical treatment of conic sections is, I am informed, a splendid monument of genius, which would still be the basis of modern study had not the treatment of these figures by analysis entirely superseded the geometrical method. Then there is Pappus in the second century A.D., who gives us in eight books a review of all the previous masters, with important additions of his own. The third name is Diophantus, who lived much later, perhaps in the fourth century, and whose work is considered the first great step toward the science of Algebra.

All these speculations were developed in the direction of mathematical physics by Archimedes, Heron, and other great men of the Alexandrian school. The triumphs of Archimedes in mechanics astonished the Romans, who, in the defence of Syracuse against their attack, found him equal to a host. But how little Archimedes confined himself to practical problems is shown by his famous method of determining the area of a circle by approximation, by inscribing and circumscribing polygons of a great number of sides, which can of course be treated and measured as a complex of triangles. This is still, I am told, the proof admitted by modern mathematicians as the best.

The works of Heron show not only an excellent practical knowledge of mechanics, but of hydrostatics, from which he deduces a number of most ingenious inventions, such as our penny in the slot, and even the construction of a whole scene acted by marionettes moving by a most elaborate hidden machinery. It[35] is a fine specimen of his ingenuity in using the ordinary mechanical contrivances. He postulates a tall hollow basis, adorned with pilasters, and having an architrave, with boards covering its upper surface. Over this stands a little round temple, visible from all sides, with six pillars. It is covered with a conical roof, and on the apex is a figure of Victory with outspread wings and holding in her right hand a garland. Under the centre of the roof stands a figure of Bacchus, holding a thyrsus in his left hand, and a cup in his right. At his feet lies a little stuffed panther. Before and behind Bacchus, and outside the temple, stands an altar with dry shavings of wood. Also on each side, outside the temple, a Bacchante, in a proper costume and attitude. The whole concern being set up at some suitable spot, the exhibitor will retire, and the automatic machine will presently move forward to a fixed spot. The moment it stops, the altar fire in front of Bacchus will light up, and from his thyrsus will flow milk or water, and from his cup wine will be poured out on the panther beneath him, the pilasters beneath will be adorned with garlands, the Bacchantes will dance round the temple; drums and cymbals will be heard. When this noise ceases, the figure of Bacchus will turn round to the other altar and all the movements be repeated in the other direction. As soon as this has happened the second time, the show is over, and the whole machine will return to its original place. We have felt bound, he adds, to make the measurements (which he gives) small, for if made large, the suspicion naturally arises in the audience that there is a man inside the machine producing all the movements. This precaution, then, should be observed in making any automatic machine.

He then proceeds to give in great detail the construction of this machine. It is as ingenious as any construction of the present day, but cannot be presented to you without a series of figures, which are given in his book. Any of you may read it in the Greek (Teubner text), to which is added an excellent German translation. It will be enough to mention that the lighting of the altar fires is done by concealing a lamp inside the altar immediately under the wood, and by withdrawing a metal plate which separates them. The flowing of milk and wine is produced by concealing two little reservoirs in the summit of the building, and leading the liquor by pipes down the inside of the pillars, and up the inside of the figure of Bacchus, so that, when the cocks are turned by machinery, the milk and wine flow and rise to the level of the thyrsus and the cup, which are set underneath the level of the cisterns. It is evident enough that people who could do these things were capable of inventing the _sakia_ now in use throughout Egypt, where a horizontal wheel worked round a capstan by oxen moves another set perpendicularly, at right angles to it, furnished with jars, which get filled below and, when they pass over the highest point of their revolution, are emptied into a water course, and so irrigate a higher level. This is well known to have been the invention of these Alexandrian mechanicians, whose theory had long preceded their practice, and whose applications of science they never valued so highly as their pure speculations.

Perhaps before leaving the subject I should tell you what was the moving force in the automatic machinery. It was a weight suspended in the air by a rope over a pulley, which, as soon as it was allowed to sink from its support, made the rope, wrapped round the axle of a large wheel, move the wheel, that was in its turn connected with other wheels. With very great and ingenious contrivance, as the machinery was all carefully concealed, the exhibitor could take his seat among the spectators, and make the ignorant believe that the whole effect was produced by some magic.